This procedure is very powerful and can be generalized at any PN order (see [1]} for other pedagogical examples of PN calculations in the near zone).
The binary dynamics in the near zone at 2PN order in the EFT framework has been computed in [2]}. It has been extended to 3PN order in [3]}, and at 4PN in [4]}, [5]}, and at 5PN in [6]}, [7]}. An alternative derivation up to 6PN can be found in [8]}, [9]}, [10]}, [11]}, [12]}, [13]}.
